Analele Universitǎţii Bucureşti, Matematicǎ
Anul LVII(2008), pp. 113–130
Sequences of Partial Defined Functions
Alexandru MIHAIL
Abstract - The aim of the paper is to study the convergence of a sequence of
partial defined functions and to study the properties of the limit function. To do
this we will define and study a new type of convergence namely local convergence
and uniform local convergence. The paper is divided in four parts. The first
part is the introduction. In the second part we compare local convergence and
uniform local convergence with simple and uniform convergence by studying the
local convergence for functions with the same domain of definition. The third part
contains the main results concerning local convergence. We will also compare local
convergence with the convergence of the graphics of the functions in the HausdorffPompeiu semidistance on the product space between the domain and codomain
spaces of functions. The last part contains sufficient conditions for the uniform
local convergence of partially defined functions.
Key words and phrases : Hausdorff-Pompeiu semidistance, partial defined function, local convergence, uniform local convergence
Mathematics Subject Classification (2000) : 28A80
1
Introduction
We start by a short presentation of Hausdorff-Pompeiu semidistance. We
will also set the notations.
For a set X, P (X) denotes the subsets of X and P ∗ (X) denotes
def
P (X) − {∅}. For a set A ⊆ P (X), A∗ = A − {∅}.
Let (X, dX ) and (Y, dY ) be two metric spaces. By C(X, Y ) we will
understand the set of continuous functions from X to Y . On C(X, Y ) we will
consider the generalized metric